Conformal weldings of random surfaces: SLE and the quantum gravity zipper

By Scott Sheffield. 2010

301 citations recorded by the source corpus.

We construct a conformal welding of two Liouville quantum gravity random surfaces and show that the interface between them is a random fractal curve called the Schramm-Loewner evolution (SLE), thereby resolving a variant of a conjecture of Peter Jones. We also demonstrate some surprising symmetries of this construction, which are consistent with the belief that (path decorated) random planar maps have (SLE-decorated) Liouville quantum gravity as a scaling limit. We present several precise conjectures and open questions.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/316d518f6915b9e7b142a875f667111976b3a005

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Citations in the corpus

Liouville quantum gravity and the Brownian map II: Geodesics and continuity of the embedding (2016, background): “We remark that the results of the current series of articles build on a large volume of prior work by the authors and others on imaginary geometry [MS12a, MS12b, MS12c, MS13a], conformal welding [She15], conformal loop ensembles [She09, SW12], and the mating of trees in infinite and finite volume se”
Liouville quantum gravity as a mating of trees (2014, methodology): “This is analogous to one of the main results of [She10] (where the notion of time is given by γ-LQG length).”
Imaginary geometry IV: interior rays, whole-plane reversibility, and space-filling trees (2013, background): “A brief overview of imaginary geometry (as defined for general functions h) appears in [26], where the rays are interpreted as geodesics of an “imaginary” variant of the Levi-Civita connection associated with Liouville quantum gravity.”
Uniqueness of critical Gaussian chaos (2017, background): “During the last 10 years there has been a new wave of interest on multiplicative chaos, due to e.g. its important connections to Stochastic Loewner Evolution [3, 29, 15], quantum field theories and quantum gravity [18, 13, 14, 24, 6, 23], models in finance and turbulence [25, Section 5], and the sta”
The Riemann zeta function and Gaussian multiplicative chaos: Statistics on the critical line (2016, background): “During the last 15 years there has been a new wave of interest in multiplicative chaos, partly due to its important connections to Stochastic Loewner Evolution [78, 4], to quantum gravity and scaling limits of random planar maps [32, 60, 13, 61, 62, 63, 27], as well as to models in finance and turbu”
Uniqueness and universality of the Brownian map (2011, background): “The very recent paper [27] contains several conjectures in this direction.”